Question

Some types of cancer are much more common than others. Although genetic and environmental factors contributing to cancer often make the news, cancer may also arise from random mutations during routine stem cell divisions over the course of a lifetime. Researchers examined the relationship between the total number of stem cell divisions in the lifetime of a given tissue and the lifetime risk of cancer in that tissue. The table provides the data for 3131 types of cancers for which this information is known in the U.S. population. Each risk value reflects the proportion of individuals in the population who get this particular cancer over their lifetime.

Lifetime risk and number of stem divisions for 3131 cancer types

?????? ????Cancer type	???????? ????Lifetime risk	Stem cell divisions	Cancer type	???????? ????Lifetime risk	???? ???? ?????????Stem cell divisions
AM leukemiaAM leukemia	0.00410000.0041000	129,900,000,000129,900,000,000	Lung (smokers)	0.08100000.0810000	9,272,000,0009,272,000,000
Basal cell	0.30000000.3000000	3,550,000,000,0003,550,000,000,000	Medulloblastoma	0.00011000.0001100	272,000,000272,000,000
CL leukemia	0.00520000.0052000	129,900,000,000129,900,000,000	Melanoma	0.02030000.0203000	763,800,000,000763,800,000,000
Colorectal	0.04800000.0480000	1,168,000,000,0001,168,000,000,000	Osteosarcoma	0.00035000.0003500	29,260,00029,260,000
FAP colorectal	1.00000001.0000000	1,168,000,000,0001,168,000,000,000	Arms osteosarcoma	0.00004000.0000400	4,550,0004,550,000
Lynch colorectal	0.50000000.5000000	1,168,000,000,0001,168,000,000,000	Head osteosarcoma	0.00003020.0000302	6,020,0006,020,000
Duodenum	0.00030000.0003000	7,796,000,0007,796,000,000	Legs osteosarcoma	0.00022000.0002200	11,130,00011,130,000
FAP duodenum	0.03500000.0350000	7,796,000,0007,796,000,000	Pelvis osteosarcoma	0.00003000.0000300	3,150,0003,150,000
Esophageal	0.00193800.0019380	1,203,000,0001,203,000,000	Ovarian germ cell	0.00041100.0004110	22,000,00022,000,000
Gallbladder	0.00280000.0028000	78,400,00078,400,000	Pancreatic ductal	0.01358900.0135890	342,800,000,000342,800,000,000
Glioblastoma	0.00219000.0021900	270,000,000270,000,000	Pancreatic islet	0.00019400.0001940	6,068,000,0006,068,000,000
Head and neck	0.01380000.0138000	31,860,000,00031,860,000,000	Small intestine	0.00070000.0007000	292,200,000,000292,200,000,000
HPV head and neck	0.07935000.0793500	31,860,000,00031,860,000,000	Testicular	0.00370000.0037000	3,348,000,0003,348,000,000
Hepatocellular	0.00710000.0071000	270,900,000,000270,900,000,000	Thyroid follicular	0.01026000.0102600	585,000,000585,000,000
HCV hepatocellular	0.07100000.0710000	270,900,000,000270,900,000,000	Thyroid medullary	0.00032400.0003240	58,500,00058,500,000
Lung (nonsmokers)	0.00450000.0045000	9,272,000,0009,272,000,000

To access the complete data set, click the link for your preferred software format:

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(a) Using the software of your choice, make a scatterplot showing lifetime cancer risk as a function of total number of stem cell divisions.

Choose the best description of the form, direction and strength of the relationship. Would it be appropriate to compute the correlation ?r between these two variables?

a. The scatterplot is not linear and no apparent correlation is observed. Because the nonlinear nature of the plot it is not appropriate to compute the correlation ?r .

b. The scatterplot is not linear and no apparent correlation is observed. Because the nonlinear nature of the plot it is appropriate to compute the correlation ?r .

c. The scatterplot is linear so it is appropriate to compute the correlation ?r .

d. The scatterplot is linear so it is not appropriate to compute the correlation ?r .

(b) When a variable spreads over several orders of magnitude, transforming the data using a base‑1010 logarithm function helps focus on these differences of magnitude. What are the minimum and maximum for each variable? Now obtain the base‑1010 log of each value in the table. What are the minimums and maximums now?

(Enter your answers rounded to four decimal places, except for maximum for lifetime risk base 10maximum for lifetime risk base 10 , which should be rounded to five decimal places.

minimum for lifetime risk base 10minimum for lifetime risk base 10: __________

maximum for lifetime risk base 10maximum for lifetime risk base 10: __________

minimum for stem cell divisions base 10minimum for stem cell divisions base 10: _________

maximum for stem cell divisions base 10maximum for stem cell divisions base 10: _________

(c) Using the software of your choice, make a scatterplot showing the log of lifetime cancer risk as a function of the log of total number of stem cells divisions. Describe the form, direction, and strength of this log–log relationship.

How strong is the relationship between the log of cancer risk and the log of stem cell division?

a. The log–log relationship of Lifetime risk and Stem cell divisions shown is negative, linear and strong.

b. The log–log relationship of Lifetime risk and Stem cell divisions shown is negative, nonlinear and weak.

c. The log–log relationship of Lifetime risk and Stem cell divisions shown is positive, linear and strong.

d. The log–log relationship of Lifetime risk and Stem cell divisions shown is positive, linear and weak.

Use software to obtain the value of ?r for the log‑transformed data. (Enter your answer rounded to four decimal places.)

?= ________

Answer 1

a) Choose the best description of the form, direction and strength of the relationship. Would it be appropriate to compute the correlation ?r between these two variables?

The scatter plot is linear so it is appropriate to compute the correlation ?r .

Option - C

b) When a variable spreads over several orders of magnitude, transforming the data using a base‑1010 logarithm function helps focus on these differences of magnitude. What are the minimum and maximum for each variable? Now obtain the base‑1010 log of each value in the table. What are the minimums and maximums now?

minimum for lifetime risk base 10: 0.00003

maximum for lifetime risk base 10: 0.50000

minimum for stem cell divisions base 10: 3,150,000

maximum for stem cell divisions base 10: 3,550,000,000,000

c) How strong is the relationship between the log of cancer risk and the log of stem cell division?

The log–log relationship of Lifetime risk and Stem cell divisions shown is positive, linear and strong.

Option - C

Use software to obtain the value of ?r for the log‑transformed data.

r = 0.0342